Math for a geodesic sphere

Question

I'm trying to create a very specific geodesic tessellation, but I can't find anything online about it.

It is normal to subdivide the triangles of an icosahedron into triangle patches and project them onto the sphere. However, I noticed an animated GIF on the Wikipedia entry for Geodesic Domes that appears not to follow this scheme. Geodesic spheres generally comprise a mixture of mostly hexagonal triangle patches, with pentagonal patches forming at the vertices of the original icosahedron; in most cases, these pentagons are linked together; that is, following a straight edge from the center of one pentagon leads to the center of another pentagon. In the Wikipedia animation, however, the edge from the center of one pentagon doesn't appear to intersect the center of an adjacent pentagons; instead it intersects the side of the other pentagon. Hopefully the drawing below makes this clear:

Where can I go to learn about the math behind this particular geometry? Ideally, I'd like to know of an algorithm for generating such tessellations.

EDIT: The following picture illustrates the more conventional scheme, whereby the centers of adjacent pentagons are linked by edges:

Answer 1

I believe it is actually just a matter of resolution (i.e., number of sub-divisions). The tessellation you show does seem to emanate from an icosahedron scheme: cf p.7 here, mid-page example. Check out the rest of the document for some calculation details - also its cited references, and some further code samples here.

Answer 2

Here's an image from one of Joe Clinton's NASA publications: